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6
УДК 517.983.24
A Class of Toeplitz Operators in Several Complex Variables
Dmitrii P. Fedchenko*
Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041
Russia
Received 10.10.2016, received in revised form 14.11.2016, accepted 20.02.2017 In order to study the Toeplitz algebras related to a Dirac operators in a neighborhood of a closed bounded domain D with smooth boundary in Cn we introduce a singular Cauchy type integral. We compute its principal symbol, thus initiating the index theory.
Keywords: Dirac operators, Cauchy type integral, symbol, Toeplitz operators, index. DOI: 10.17516/1997-1397-2017-10-2-206-215.
There are a number of ways in which the theory of Toeplitz operators can be generalised to n dimensions, see e.g. [1,2] and the references given there. The monograph [3] presents much more advanced theory of Toeplitz operators in several complex variables.
The paper [4] describes precisely how Toeplitz operators of "Bergman type" are related to Toeplitz operators of "Szego type". A remarkable connection between the theory of Toeplitz operators a la [1] and the standard theory of pseudodifferential operators emerged from the work [5]. This connection in its broad outlines is elucidated in [4], too.
This work focuses on a new class of Toeplitz operators which is more closely related to elliptic theory. The new Toeplitz operators admit very transparent description which motivates strikingly their study. To this end, let A be a (kxk) -matrix of first order scalar partial differential operators in a neighborhood of the closed bounded domain D with smooth boundary S in C". Assume that the leading symbol of A has rank k away from the zero section of the cotangent bundle of D. Then, given any solution u of the homogeneous equation Au = 0 in the interior of D which has finite order of growth at the boundary, the Cauchy data t(u) of u with respect to A possess weak limit values at the boundary. If A satisfies the so-called uniqueness condition of the local Cauchy problem in a neighborhood of D, then the solution u is uniquely defined by its Cauchy data at S. Let t(u) = Bu be a representation of the Cauchy data of u. The space of all Cauchy data Bu of u at the boundary is effectively described by the condition of orthogonality to solutions of the formal adjoint equation A* g = 0 near D by means of a Green formula, see [6, § 10.3.4]. In this way we distinguish Hilbert space of vector-valued functions on S which represent solutions to Au = 0 in the interior of D. In particular, one introduces Hardy spaces H as subspaces of L2(S, Ck) consisting of the Cauchy data of solutions to Au = 0 in the interior of D with appropriate behaviour at the boundary. Pick such a Hilbert space H. By the above, H is a closed subspace of L2(S, Ck) and we write П for the orthogonal projection of L2(S, Ck) onto H. If A is the Cauchy-Riemann operator then П just amounts to the Szego projection.
Given a (k x k) -matrix M(z) of smooth function on S, the operator TM on H given by u ^ n(Mu) is said to be a Toeplitz operator with multiplier M. If M is a scalar multiple of the
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unit matrix, then the theory of Toeplitz operators TM is much about the same as the classical theory of Toeplitz operators. Otherwise the theory is much more complicated.
Finally, we briefly discuss Toeplitz operators related to the algebra of octonions O introduced by John T. Graves in 1843. The octonions were discovered independently by Cayley and are sometimes referred to as Cayley numbers or the Cayley algebra.
1. Dirac operators
Let Cn be the standard n-dimensional complex space obtained from the underlying real space R2n of variables x — (x1,... ,x2n) by introducing the complex structure
Zj — xj + lxn+j
for j — 1,... ,n. As usual, we define complex derivatives by
d 1 ( d d — i-
dzj 2 \0xj dxn+j
d 1 ( d d
+1-
dzj 2 y dxj dxn+j Let us consider a matrix-valued constant coefficient operator
such that
^ M) = g "j +ßj Wj (1)
= - 1AE (2)
4
where A* is the formal adjoint of A and E the identity matrix. It is easily seen that the identity (2) reduces to a system of identities for the coefficients, namely
a*ak + /3*ftj — 5jkE,
(3)
a* 3k + a *k 3j — 0
for all j,k — 1,... ,n.
It is well known that there is a solution of (3) amongst matrices of type (2n-1 x 2"-1), cf. for instance Chapter 3 in [7].
First-order differential operators with constant coefficients factorizing the Laplacian in the sense of (2) are called Dirac operators.
Example 1.1. The Cauchy-Riemann operator A — d/dz is a Dirac operator in the complex plane.
Example 1.2. The operator
is a Dirac operator in C2.
U = ^ d/dzi -3/3z2
d/dz2 d/dz\
)
2. Fundamental solution
To construct an explicit fundamental solution $ for a Dirac operator A we make use of relation (2). Namely, denote by e the standard fundamental solution of convolution type for the Laplace operator in R2". In the coordinates of C" it reads
(n - 1)! 1 1
e(z) =
2nn 2 - 2n \z\2n-2'
if n> 1, and e(z) = (1/2n) ln \z\, if n = 1. Set
d ( , (n - 1)! A*(z - Z,z - Z)
dCje Z ) nn \z - Z\2n
$(z - Z)=4A* (KdZ,d^)e(z - Z) =
for z = Z, where by A*(z — Z,z — Z) is meant the adjoint of the matrix A(z — Z,z — Z).
Lemma 2.1. As defined above, $(z — Z) is a fundamental solution of the Dirac operator A, i.e. $ oA = E and Ao $ = E on C^mp(Cn, C), where (k x k) is the type of E.
Proof. The first relation $ o A = E is fulfilled by the very construction of $. Since A is a square matrix, if follows from (2) that AA* = — (1/4)AE whence $* oA* = E on C^^C"*, Ck). The latter equality is equivalent to Ao $ = E, as desired. □
3. Green formula
Let D be a bounded domain with smooth boundary S = dD in Cn. Write v(y) = (v\(y),..., v2n(y)) for the unit outward normal vector of S at a point y € S. If p(x) is a defining function of S then
( ) Vp(y)
V (y) =
\Vp(y)\
for y € S, where Vp(y) stands for the real gradient of p at y. The complex vector vc = (vCj1,..., vc,n) with coordinates vCj = Vj + ivn+j is called the complex normal of the hypersurface S. In the coordinates of Cn we obviously have
dp/dZj
№ P(Z )\
for j = 1,... ,n.
Denote by dZ the wedge product dZ1 A •• • A dZn, and by dZ[j] the wedge product of all dZi, • • •,dZn but dZj.
Lemma 3.1. For each j = 1,... ,n, the pullback of the differential form dZ A dZ [j] under the embedding S ^ Cn is equal to (-l)j-1(2i)n-1 ivCjds, where ds is the area form on the hypersurface S induced by the Hermitian metric of Cn.
Proof. An easy computation shows that the pullback of the differential form dy[j] under the embedding S ^ Cn is equal to (-l)j-1vjds, for every j = 1,..., 2n. From this the lemma follows immediately. □
v
Denote by a(Z) the principal homogeneous symbol a1(A)(C, 0 of the operator A evaluated at the point (Z, vc(Z)/i) of the cotangent bundle of a neighborhood of D, where Z G S. For example, the principal homogeneous symbol of the Cauchy-Riemann operator is equal — (1/2)(^1 + i£2).
We are now in a position to specify the restriction of the Green operator G^(g,u) of A to the boundary. By a Green operator of A is meant a bilinear operator Ga from CTO(Cn, (Ck)*) x C~(Cn,Ck) to differential forms of degree 2n — 1 on Cn, such that dGA(g*,u) — ((Au,g)y — (u, A*g)y)dy holds pointwise in Cn.
By [8, § 2.4.2], there is a unique Green operator for A, and its pullback under the embedding S Cn is
1 i i G^(g,u) — ~^gA [^(vj — iVn+j), 2(vj + iVn+j)J uds — ga(Z)uds
whence _
Ga($(Z — Z), E) — ^^ ^Z) a(Z)ds. (4)
Lemma 3.2. Every vector-valued function u G C 1(D, Ck) has the integral representation Xv u — — GA(Q(z — .),u)+ <b(z — .)Audv,
Js Jv
where dv is the Lebesgue measure in R2n and xv the characteristic function of D.
Proof. This is a very special case of a general Green formula related to an elliptic system of differential equations, see for instance [8, § 2.5.4] and elsewhere. □
Needless to say that this formula extends to the case of Sobolev class functions u G H 1(D, Ck) as well as to more general functions on D.
4. Toeplitz operators
Theorem 4.1. Let u0 G L2(S, Ck). In order that there be a solution u to Au — 0 in the interior of D, which has finite order of growth at S and coincides with u0 on S, it is necessary and sufficient that
/ ga(Z)uo ds — 0 (5)
Js
for all solutions of the formal adjoint equation A*g — 0 near D.
Proof. See Theorem 10.3.14 in [6]. □
We denote by H the (closed) subspace of L2(S, Ck) that consists of all functions u satisfying the orthogonality conditions (5). The elements of H can be actually specified as solutions to Au — 0 of Hardy class H2 in the interior of D, see [6, § 11.2.2]. The orthogonal projection n of L2(S, Ck) onto H is therefore an analogue of Szego projection.
Definition 4.2. Let M(z) be a (k x k)-matrix whose entries are bounded functions on S. By a Toeplitz operator TM with multiplier M is meant the operator u ^ n(Mu) in H.
More generally, if ^ is a (k x k)-matrix of pseudodifferential operators of order 0 on S, then ^ maps L2(S, Ck) continuously into L2(S, Ck). Therefore, the composition T$ — n^ is a continuous self-mapping of H which we call a generalised Toeplitz operator.
If the projection n is a classical pseudodifferential operator on S, then from the equality n2 — n it follows readily that the order of n just amounts to zero. Hence, the generalised Toeplitz operators on S form a subalgebra of the C*-algebra of all zero order classical pseudodifferential operators on C(X'(S, Ck).
5. The generalised Cauchy type integral
To clarify the nature of the generalised Szego projection n we introduce the singular Cauchy type integral
Cu(z) = — p.v. / CA($(z — .),u) (6)
JS
for z € S, where u € L2(S, Ck). The principal value of the integral on the right-hand side exists for almost all z € S and it induces a bounded linear operator in L2(S, Ck).
Lemma 5.1. The operators (1/2)I ±C are projections on the space L2(S, Ck).
Proof. This follows from the equality C2 = (1/4)I by a trivial verification, cf. for instance [9]. □ The generalised Cauchy type integral (6) is a classical pseudodifferential operator of order 0 in C(S, Ck). We finish this section by evaluating its principal homogeneous symbol. To this end, we identify the cotangent space T*S of S at a point z € S with all linear forms on T*R2n which vanish on the one-dimensional subspace of T*R2n spanned by v(z). Since T**R2n = R2n, one can actually specify T*S as the hyperplane through the origin in R2n which is orthogonal to the vector v(z).
Lemma 5.2. For each z € S and £ € T*S, the symbol of order 0 of the operator C is equal to
a0(C)(z,£) = a1(A*^z, a(z).
Proof. By assumption, the Laplacian A* A is a second order elliptic differential operator on Ctt(U, Ck). It has a parametrix P which is a (k x k)-matrix of scalar pseudodifferential operators of order —2 on U. The operator $ differs from PA* by a smoothing operator, and so it has the principal homogeneous symbol (a2(A*A))-1 a1 (A*) which is a left inverse for a1(A). Formula (6) just amounts to saying that
C u(z) = —$(a(Z )u(Z )ls),
where ls is the surface layer on S. We thus see that the pseudodifferential operator C on S is the restriction to S of the pseudodifferential operator
V = —$ o a(z)
defined in a neighborhood of S. This latter is of order — 1 and its principal symbol is easily evaluated, namely
a-1(V)(z,£) = — a-1($)(z,£) a(z) = —(a2(A*A)(z,£))-1 a(z)
for z in a neighborhood of S and £ € Cn \ {0}. A familiar argument now shows that the principal symbol of C is given by the formula
1 f
a0(C)(z,£) = — p.v. J a-1 (V)(z,tv(z) + £) dt =
= — ^p.v.r (a2(A*A)(z,tv(z)+ £))-1 a1(A*)(z,tv(z) + £) dt
for all z € S and £ € R2n \ {0} orthogonal to v(z). Note that the integral on the right-hand side diverges, however, its Cauchy principal value exists, which is due to the condition (v(z),£) = 0.
Since
we shall have established the desired equality if we prove that
1 r ~ 1 _ 1 1
4n]-^ \tv(z) + £\2 t _ 4 for all z G S and £ G M2" \ {0} orthogonal to the vector v(z). A trivial verification shows that
\tv(z)+ £\2 _ t2 + \£\2
whence
1 n 1 ,1 f^ 1,11
dt = — -—77T dt = — — ,
4n J\tv(z) + £\2 4n J_t2 + \£\2 4 \£\' as desired. □
6. Index of Toeplitz operators
We now return to generalised Toeplitz operators T* = n^ in H introduced in Section 4., see Definition 4.2 and below. Here, n is a projection of L2(S,Ck) onto H. We restrict ourselves to the case where n is a classical pseudodifferential operator of order zero in CS, Ck). We are interested in characterizing those Toeplitz operators in H which possess the Fredholm property. To this end we extend T* from H to all of L2(S, Ck) in a special manner and use Fredholm criteria for operator algebras with symbols.
Lemma 6.1. A Toeplitz operator T* in H is Fredholm if and only if so is the operator
E* := T*n +(I — n)
in L2(S, Ck).
Proof. We see that T* is the restriction of the pseudodifferential operator E* = T*n © (I — n) : H © HL ^ H © HL
on H.
If E* is Fredholm then ind E* is finite. But ind E* = indT* +ind (I — n), where T* : H ^ H and (I — n) : H^ ^ HIt is clear that ind (I — n) = 0, whence ind T* is also finite. And moreover, ind E* = ind T* . □
The operator E* on L2(S, Ck) is pseudodifferential of order zero. Its principal homogeneous symbol just amounts to a0(E*) = a0(n)a0(^)a0(n) + a0(I — n) away from the zero section of T*S. Given any s G R, the operator E* in Hs(S, Ck) is known to be Fredholm if and only if it is elliptic, i.e. a0(E*)(z,£) is invertible for all (z,£) G T*S with £ = 0. Moreover, the index of this operator is actually independent of the particular choice of s and it can be evaluated by the familiar Atiyah-Singer formula [10].
7. Concluding remarks
In the sequel we restrict our attention to a special Clifford algebra corresponding to the case n = 4. Clifford algebras have important applications in variety of fields including string theory, special relativity and quantum logic. The algebra in question is called the algebra of octonions and denoted by O. To wit,
qi -q2 B di
.4 =
i qi -q2 \ \ d qi J '
where qi
id d
V dzi dz2+i
i >)
j , j is the fundamental quaternion unit and the bar denotes the
conjugate of a quaternion.
Let us rewrite the action of the operator A on a quaternion-valued function in complex coordinates, that is
Au
di + dsj —d2 — d4j d2 — d4j di — dsj
)(
/ di —d 2 —d3C d4C \ ( Ui \
Ui + U3j \ = d2 di 4c Bsc U2
U2 + U4j ) d3C —d4C i — d2 Us
\ —d4c —dsC 2 i \ U4 )
c being the complex conjugation and d j = d/ddj.
The octonions have the dimension eight. Because of nonassociativity they can not be represented as quaternion (2 x 2)-matrices with usual multiplication. The product of two octonions O = (a, b) and P = (c, d) is defined via the Cayley-Dickson construction by
(a, b)(c, d) = (ac — db, da + be).
It corresponds to the special multiplication of the matrices
op
( a —b c —d \ = (
b a d c =
ac — d b — da — cb + ad —bd +
It remains to show that the multiplication Op is alternative, i.e. (PO)O = P(OO) for all matrices O, p.
Lemma 7.1. The matrices O constitute an alternative algebra. Proof.
bc . a
O(OP)
O(OP)
a —b b a
)(
ac — d b —da — bc cb + ad — bd + da
aac — adb — cbb — adb —daa — bea + bb d ■ acb — dbb + acb + aad — bda — b bd — b da -
bc a c aa
and
(OO)P
dc —dc
aa — bb —ba — ba c —d
ab - ab —bb - aa d c
aac — bbc — dba — dba —daa + dbb — bad
ca b - cab — bbd - aad —abd — abd — c bb -
bac caa
(OO)P and
(
where
adb + adb — dba — dba = [a, db] + [a, db] = [a + a, db] = 0
and
acb + acb — cab — cab = [a, c] b + [a, c] b = [a + a, c]b = 0.
The equality (PO)O = P(OO) is verified similarly. □
To specify the class of multipliers we describe those (2 x 2)-matrices M(z) which commute with a1(A)(z,C).
Lemma 7.2. A (2 x 2)-matrix M commutes with a1(A)(z, Z) if and only if M is of the form
M =( tc T)' (7>
C being quaternion conjugation and X, Y are arbitrary (2 x 2)-matrices with complex entries.
Proof. Here the multiplication OM is considered in the usual way. Straightforward calculation shows that
( [a, X] — [b, Y]C —[b, X] — [a, Y]C \ 1 ' J V [b,X ] + [a, Y]C [a, X ] — [b,Y]Cj-An easy computation shows that [a, X] = 0 if and only if X is of the form
X = ( W1 —W2C ) , (8)
\ W2C W1 J
where wi, w2 are arbitrary complex numbers. □
Needless to say that X is a (2 x 2)-matrix of nonlinear operators. From the equality
X1X2 =( W\WX — w1w2 —(w2w2 + w1w2 )c \
X. ^V — I . i_2 1 2 \ 12 1_2 I
y (wiw2 + wiw2)C wiw2 — wiw2 ) it follows readily that the matrices X of the form (8) constitute an unital algebra. Since det X = w2 + \w2\2, a matrix X is invertible if and only if X = 0.
Since the kernel and cokernel of a nonlinear operator X fail to be vector spaces, the index of X is no longer defined. By the index of a differentiable nonlinear operator is usually meant the index of its Frechet derivative. However, the operators in question are not Frechet differentiable in the sense of complex vector spaces.
Lemma 7.3. The corresponding operator (8) is not differentiable unless w2 = 0.
Proof. This follows by immediate computation. □
We conclude that there is no suitable Fredholm problems over the field of complex numbers for Toeplitz operators with multipliers of special kind. From now on all consideration are over the field of real numbers.
Using Lemma 5.1 we establish a very useful formula for the projection n, namely, n = (1/2)I + C. Thus,
TM i = (1/2)M 1 + CM1, TM 2 = (1/2)M2 + CM2,
and so
TmiTM2 = TMiM2 — [C, M2]((1/2)M1 — CM1). (9)
Lemma 7.4. Assume that M1 and M2 are (2 x 2)-matrices of smooth functions on S. Then TM2m 1 = TM2 TMi modulo compact operators on H.
Proof. If a (2 x 2)-matrix M commutes with a1(A)(z,£), it is of the form (7), and so the adjoint matrix M* has the same form. Hence it follows that M* commutes with a1(A)(z, £), and so M commutes with the adjoin a1(A*)(z,£), too. Now an elementary analysis shows readily that
C(Mu)(z) = -p.v. f M(Z)$(z - Z)a(Z)u(Z)ds =
J S
= M(Cu)(z)+ f aA($(z - Z), (M(z) - M(Z))u)z
S
holds almost everywhere on the boundary for all u G L2(S, C4). In particular, if u G H, then
C(Mu)(z) = (l/2)(Mu)(z)+ f aA($(z - Z), (M(z) - M(Z))u)z
S
for almost all z G S. From these two equalities we conclude that the remainder [C, M2]((1/2)M1 -CM^ in (9) is a pseudodifferential operator of order -2 on the surface S. □
One deduces from the proof of Lemma 7.4 that TM2Ml = TM2TMl holds actually up to trace class operators, if n = 1. In this case the results of [11] apply to evaluate the index of Fredholm Toeplitz operators.
Lemma 7.4 allows one to develop the Fredholm theory of Toeplitz operators with operator-valued multipliers of the form (7).
The research of the author was supported by the Deutscher Akademischer Austauschdienst and by the Russian Federation President grant of support of leading scientific schools NSH-9149.2016.1.
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Об одном классе теплицевых операторов в случае многих комплексных переменных
Дмитрий П. Федченко
Институт математики и фундаментальной информатики Сибирский федеральный университет Свободный, 79, Красноярск, 660041
Россия
Чтобы изучать теплицевы алгебры, связанные с операторами Дирака в окрестности замкнутой ограниченной области D с гладкой границей в Cn, мы 'рассматриваем сингулярный интеграл типа Коши. Мы вычисляем его главный символ, делая тем самым первый шаг к построению формул индекса для операторов Тёплица.
Ключевые слова: операторы Дирака, интеграл типа Коши, символ, теплицевы операторы, индекс.
